Doceamus
doceamus . . . let us teach
Problem Solving:
Moving from Routine to
Nonroutine and Beyond
Barry Garelick
An important part of the job of teaching math in
K–12 is to stretch students—to teach them creative
and personal engagement with the material. At
some point this must involve expecting students to
come up with previously unfamiliar steps on their
own for new problems that do not lend themselves
to known algorithms, prescribed methods, and predictable approaches. An effective way of doing this
is to extend routine problems that students know
how to solve into nonroutine problems.
Over the past two decades, however, disagreements between advocates of traditional or conventional math teaching and the math reform
movement have resulted in a fragmented approach
to teaching math. A key area of disagreement
centers on the distinction between “exercises”
and “problems”. Math reformers generally believe
that conventional math teaching consists mainly
of routine problems that are nonthinking, repetitive, tedious and do not lead to students learning
to solve nonroutine problems.
Barry Garelick is a high school mathematics teacher in
California. He majored in mathematics at the University
of Michigan. His email address is barryg99@yahoo.com.
Members of the Editorial Board for Doceamus are: David
Bressoud, Roger Howe, Karen King, William McCallum,
and Mark Saul.
DOI: http://dx.doi.org/10.1090/noti1050
1340 Notices
of the
One math reform approach has been to
present students with a steady diet of “challenging problems” that neither connect with
the students’ lessons and instruction nor develop any identifiable or transferrable skills.
The following problem from Hjalmarson and
Diefes-Dux (2008) is one example: How many boxes
would be needed to pack and ship one million
books collected in a school-based book drive? In
this problem the size of the books is unknown
and varied, and the size of the boxes is not stated.
While some teachers consider the open-ended nature of the problem to be deep, rich, and unique,
students will generally lack the skills required to
solve such a problem, skills such as knowledge of
proper experimental approaches, systematic and
random errors, organizational skills, and validation and verification.
Based on my experiences as both student and
teacher, as well as the experiences of veteran math
teachers, I submit that a substantial education in
mathematics should steer a middle course between
the proliferation of routine problems and reliance
upon unique, complex projects. Students should
learn to apply basic principles in a much wider
variety of situations than typically presented in
texts. Such problems, however, should not be as
complex or as time consuming as the example
above. A math problem is not necessarily useful
just because it requires outside-of-the-box insight
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Volume 60, Number 10
and/or inspiration and will generally not result in
a problem-solving “habit of mind”.
An airplane traveling 280 miles per
hour leaves San Francisco 13½ hours
after a steamship has sailed. If the
plane overtakes the ship in 1½ hours,
find the rate of the steamship.
Extending Routine Algorithmic Problems
to Nonroutine Problems
Problems that are extensions of routine problems,
(as suggested by Henderson and Pingry (1953))
can effectively build upon prior knowledge and
develop a larger problem-solving repertoire. Such
extensions require students to synthesize ideas
and procedures from two or more areas of learned
procedures and are frequently multistep. For example, while most students have learned how to
factor a2 – b 2, they are likely to find the following
difficult to factor: a2 – x2 + 2bx – b 2. This problem requires students to synthesize what they
know about perfect square trinomials, how they
are factored, and how these ultimately relate to
factoring the difference between two squares. An
intermediate problem may be introduced, such as
a2 – (m – n)2, to serve as scaffolding for the more
challenging version.
The factoring of the difference of two squares
can be extended even further: Find two positive
integers “a” and “b” such that a2 – b 2 = 2,011. A
student may first factor to obtain (a + b)(a – b) =
2,011. Some students may not be able to go any
further. Others will see that expressing the righthand side of the equation as the product of two
factors xy puts them on the pathway to solving it.
Students may be stumped and will overlook the
fact that 2,011 and 1 can represent x and y.
While the variations between these two problems are slight (one asks for time traveled, the
other speed), they present a challenge to beginning students. Varying the structure of the basic
problem forces the student to read each problem
carefully. They must identify what the question
is, what data are provided, and how to transform
it into an equation. The variation and difficulty
increases so that the last problems of the set are
more complex and nonroutine:
A ship must average 22 miles per hour
to make its ten-hour run on time. During the first four hours, bad weather
caused it to reduce speed to 16 miles
per hour. What should its speed be for
the rest of the trip to keep the ship to
its schedule?
Extending the nonroutine distance problem
even further, students can be given the following
problem in an Algebra 2 course:
Two boys in a canoe are at a confluence
of a lake and a river. A house upriver
and a boulder on the other side of the
lake are equidistant from the boys.
Would it take the same amount of time
to paddle across the lake to the boulder
and back as it would to paddle to the
house upriver and back (at the same
speed relative to the water each way
for both options)? If not, which option
would take longer?
Extending Routine Word Problems
to Nonroutine Problems
A frequent criticism of word problems in textbooks
is that they present a worked example/method for
solving a particular type of problem, followed by a
set of almost identical problems to solve. Students
may experience the practice of applying a memorized technique and mechanically look for the data
to plug in to the appropriate equations without
having to read the problem. But what if you are
given values with different units in a distance/rate
problem? And what if you are given two legs and
two different rates and need to find your average
rate? Word problems can and should be varied for
improving problem-solving ability.
This is, in fact, what is done in well-written
algebra textbooks or in problem sets devised by
teachers. Students are given instruction via worked
examples and some initial practice problems. After
that, the problems vary. For example, consider
the following two distance/rate problems from
Dolciani et al. (1962):
A freight train left Beeville at 5 AM at
30 miles per hour. At 7 AM an express
train traveling 50 miles per hour left
the same station. When did the express
overtake the freight?
November 2013
This problem does not allow a “plug-in” solution. Students must model the situation using symbols and apply their knowledge of expressing time
as a function of rate and distance. Many students
will assume that either route will take the same
amount of time, reasoning that the amount that the
canoe is slowed down by the current when traveling upstream is canceled by the additional speed
the current imparts when traveling downstream.
Proving or disproving the initial intuitive response,
however, requires mathematics. Ultimately, it involves an inequality that does not lend itself to an
algorithmic solution.
Building a Problem-Solving Repertoire
Students should routinely be presented with
nonroutine problems. Even if a student fails to
solve such a problem, seeing its solution opens
up pathways in which they discover relationships
between content and what they know about solving
routine problems.
Notices
of the
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Sweller et al. (2010) state that problem solving
cannot be taught independently of basic tools and
basic thinking. Over time, students build up a repertoire of problem-solving techniques. Ultimately,
the difference between someone who is good
and someone who is bad at solving nonroutine
problems is not that the good problem solver has
learned to solve novel, previously unseen problems. It is more the case that, as students increase
their expertise, more nonroutine problems appear
to them as routine.
ematics: Its Theory and Practice (H. F. Fehr, (ed.),
Washington, D.C., National Council of Teachers of
Mathematics, pp. 228–370.
References
Mary Dolciani, S. L. Berman, and J. Freilich
(1962), Modern Algebra, Book 1, Houghton Mifflin
Co. Boston, MA.
John Sweller, R. Clark, and P. Kirschner
(2010), Teaching general problem-solving skills
is not a substitute for, or a viable addition to,
teaching mathematics, Notices of the American
Mathematical Society, Vol. 57, No. 10, November
2010. http://www.ams.org/notices/201010/
rtx101001303p.pdf
Ken B. Henderson and R. E. Pingry (1953), Problem solving in mathematics, The Learning of Math-
Margret A. Hjalmarson and Heidi Diefes-Dux
(2008), Teacher as designer: A framework for
teacher analysis of mathematical model-eliciting
activities, Interdisciplinary Journal of Problembased Learning, Vol. 2, Iss. 1, Article 5. Available at http://dx.doi.org/10.7771/1541–
5015.1051.
In Memory of Professor
Francisco Federico Raggi
Cárdenas
María José Arroyo, Rogelio Fernández-Alonso González, Sergio R.
López-Permouth, José Ríos Montes, and Carlos Signoret
Professor Francisco Federico Raggi Cárdenas
passed away June 12, 2012, in Mexico City. Professor Raggi was born in Mexico City in 1940. He obtained his undergraduate degree from Universidad
Nacional Autónoma de México (UNAM), a master’s
degree from Harvard, and a doctorate from UNAM.
In 1962 he joined the Institute of Mathematics
of UNAM, where he worked until his passing. He
held visiting positions in universities throughout
the world, including institutions in Germany, the
United States, Belgium, Canada, Spain, and Italy.
Professor Raggi taught modern algebra at all
levels to many generations of students in Mexico.
In particular, he had four Ph.D. students. Raggi was
author or coauthor of five books that are prominently used in universities throughout Mexico. As
he traveled constantly and taught at many places
throughout Mexico, Raggi’s commitment to the
María José Arroyo is professor of mathematics at Universidad Autónoma Metropolitana Iztapalapa. Her email
address is mja@xanum.uam.mx.
Sergio R. López-Permouth is professor of mathematics at
Ohio University. His email address is lopez@ohio.edu.
Rogelio Fernández-Alonso González is professor of mathematics at Universidad Autónoma Metropolitana Iztapalapa. His email address is rojo99@prodigy.net.mx.
DOI: http://dx.doi.org/10.1090/noti1052
1342 Notices
of the
José Ríos Montes is professor of mathematics at Universidad Nacional Autónoma de México. His email address is
jrios@matem.unam.mx.
Carlos Signoret is professor of mathematics at Universidad
Autónoma Metropolitana Unidad Iztapalapa. His email
address is casi@xanum.uam.mx.
AMS
Volume 60, Number 10